Mike wrote:
> there's a general graph theory conjecture that cayley graphs are
> hamiltonian (i.e. have hamiltonian circuits).
>
> if we take the cayley graph formed by generators
> {F, F', L, L' U, U', R, R', B, B', D, D'}, the conjecture asserts
> that there is a sequence of N quarter turns that visits every
> position exactly once and returns to START.
> (here N = 43252003274489856000 is the order of the group.)

Also, I seem to remember that the slice-squared group had 8 elements,
and if you graphed a route through the elements it formed a cube.
After drawing such a graph it is not hard to find a hamiltonian
circuit (using the edges of the cube as a pathway).

This may be true in general for all the platonic solids.
(I need to re-check "Regular Polytopes" by Coxeter).

So we have 2 examples and no counter-examples of the general graph
theory Mike mentions.